1. INTRODUCTION

For the last century astronomers have been modelling the structure of
`nebulae', and here we focus on those external to the Milky Way. A key
activity performed by many astronomers, past and present, is the
catergorisation of these galaxies
(Sandage 2005)
and the quantification
of their physical properties. How big are they? How bright are they?
What characteristics distinguish or unite apparent subpopulations?
Answers to such questions, and the establishment of "scaling
relations" between two or more galactic properties provides
valuable insight into the physical mechanisms that have shaped
galaxies.

Popular scaling relations involving global galaxy parameters such as size,
surface brightness, luminosity and concentration are reviewed here. As we
shall see, many bivariate distributions, which are frequently assumed to be
linear, are often only approximately so over a restricted luminosity range.
For example, it may come as a surprise for many too learn that the useful
Kormendy (1977b)
relation is only the tangent to the bright arm of a
continuous but curved effective radius-(surface brightness) relation which
unifies dwarf and giant elliptical galaxies
(section 3.2.4). Similarly, the
Faber-Jackson (1976)
relation with a slope of 4 represents the
average slope over a restricted luminosity range to what is a curved or
broken luminosity-(velocity dispersion) distribution, in which the slope
is 2 rather than 4 at lower luminosities
(section 3.3.3).
Knowing these trends, the bulk of which cannot be established when assuming
structural homology, i.e. using
de Vaucouleurs' (1948)R1/4 model, is
vital if one is to measure, model and make sense of galaxies.

This article has been structured into four main sections.
Section 1 provides this general overview plus a further review and
introduction to galaxies on the Hubble-Jeans sequence
1. Included are diagrams
showing the location of dynamically hot stellar systems in the mass-size
and mass-density plane, revealing that some high-z compact galaxies
have properties equivalent to the bulges of local disc galaxies.
Section 2 provides an historical account of
how the radial distribution of stars in elliptical galaxies have been
modelled, and the iterative steps leading to the development of the
modern core-Sérsic model
(section 2.2). Subsections cover the
Sérsic model
(section 2.1), its relation and
applicability to dark matter halos
(section 2.1.1), partially-depleted
galaxy cores
(section 2.2.1), excess nuclear light
(section 2.3) and
excess light at large radii in the form of halos or envelopes around giant
elliptical galaxies (section 2.4).
Section 3 presents and derives a number of
elliptical galaxy scaling
relations pertaining to the main body of the galaxy. From just two linear
relations which unite the faint and bright elliptical galaxy population
(section 3.1), a number of curved
relations are derived
(section 3.2). Several broken relations,
at MB
-20.5 mag, are additionally presented in
section 3.3.
For those interested in a broader or different overview of elliptical
galaxies, some recent good reviews include
Renzini (2006),
Cecil & Rose (2007),
Ciotti (2009)
and
Lisker (2009.
Finally, the latter third of this paper is tied up in
section 4
which contains a discussion of the light
profiles of disc galaxies and their bulge-disc decomposition
(4.1).
Also included are subsections pertaining to dust
(section 4.2), the difficulties with
identifying pseudobulges (section 4.3),
potential bulgeless galaxies
(section 4.4) and methods to model bars
(section 4.5).
Throughout the article references to often overlooked discovery or pioneer
papers are provided.

Looking out into the Milky Way arced across our night sky, the notion
that we are residents within a pancake-shaped galaxy seems reasonable to
embrace. Indeed, back in 1750 Thomas Wright also conjectured that we
reside within a flat layer of stars which is gravitationally bound and
rotating about some centre of mass. However, analogous to the rings of
Saturn, he entertained the idea that the Milky Way is comprised of a
large annulus of stars rotating about a distant centre, or that we are
located in a large thin spherical shell rotating about some divine
centre (one of the galactic poles). While he had the global geometry
wrong, he was perhaps the first to speculate that faint, extended
nebulae in the heavens are distant galaxies with their own (divine)
centers.

As elucidated by
Hoskin (1970),
it was
Immanuel Kant (1755),
aware of the elliptically-shaped nebulae observed by Maupertuis, and
working from an incomplete summary of
Wright (1750)
that had been published in a Hamburg
Journal 2, who effectively
introduced the modern concept of disc-like galactic distributions of
stars - mistakenly crediting Wright for the idea.

Using his 1.83 m "Leviathan of Parsonstown" metal reflector
telescope in Ireland, Lord William Henry Parsons, the 3rd Earl of Rosse,
discovered 226 New General
Catalogue 3 (NGC:
Dreyer 1888)
and 7 Index Catalogue (IC:
Dreyer 1895, 1908)
objects
(Parsons 1878).
Important among these was his detection of spiral structure in many
galaxies, such as M51 which affectionately became known as the whirlpool
galaxy.

With the discovery that our Universe contains Doppler shifted 'nebulae'
that are expanding away from us
(de Sitter 1917;
Slipher 1917;
see also
Friedmann 1922,
Lundmark 1924
and the reviews by
Kragh & Smith 2003
and
Shaviv 2011),
in accord with a redshift-distance relation
(Lemaitre 1927,
Robertson 1928,
Humasson 1929,
Hubble 1929)
5 - i.e. awareness that
some of the "nebuale"
are external objects to our galaxy - came increased efforts to
catergorise and organise these different types of "galaxy". As noted by
Sandage (2004,
2005),
Sir James Jeans (1928)
was the first to present the (tuning fork)-shaped diagram that encapsulated
Hubble's (1926)
early-to-late type galaxy sequence, a sequence which had been inspired
in part by
Jeans (1919)
and later popularised by Hubble
(1936a;
see
Block et al. 2004).
Quantifying the physical properties of galaxies
along this sequence, with increasing accuracy and level of detail, has
occupied many astronomers since. Indeed, this review addresses aspects
related to the radial concentration of stars in the elliptical and disc
galaxies which effectively define the Hubble-Jeans sequence. Irregular
galaxies are not discussed here.

For reasons that will become apparent, this review uses the galaxy
notation of Alan Sandage and Bruno Binggeli, in which dwarf elliptical
(dE) galaxies are the faint extension of ordinary and luminous
elliptical (E) galaxies, and the dwarf spheroidal (dSph) galaxies -
prevalent in our Local Group
(Grebel 2001)
- are found at magnitudes fainter than MB -13 to -14 mag
( 108M in
stellar mass; see
Figure 1a). Figure 1a
reveals a second branch of elliptically-shaped object stretching from
the bulges of disc galaxies and compact elliptical (cE) galaxies to
ultra compact dwarf (UCD) objects
(Hilker et al. 1999;
Drinkwater et al. 2000;
Norris & Kannappan
2011
and references therein). A possible connection is based upon the
stripping of a disc galaxy's outer disc to form a cE galaxy
(Nieto 1990;
Bekki et al. 2001b;
Graham 2002;
Chilingarian et al. 2009)
and through greater stripping of the bulge to form a UCD
(Zinnecker et al. 1988;
Freeman 1990;
Bassino et al. 1994;
Bekki 2001a).
It is thought that nucleated dwarf elliptical galaxies may
also experience this stripping process, giving rise to UCDs.

While the identification of local spiral galaxies is relatively free
from debate, the situation is not so clear in regard to
elliptically-shaped galaxies. The discovery of UCDs, which have sizes
and fluxes intermediate between those of galaxies and (i) the nuclear
star clusters found at the centres of galaxies and (ii) globular
clusters (GCs: e.g.
Hasegan et al. 2005;
Brodie & Strader
2006),
led
Forbes & Kroupa
(2011)
to try and provide a modern definition for
what is a galaxy (see also
Tollerud et al. 2011).
Only a few years ago
there was something of a divide between GCs and UCDs - all of which had
sizes less than ~ 30 pc - and galaxies with sizes greater than 120 pc
(Gilmore et al. 2007).
However, as we have steadily increased our celestial inventory, objects
of an intermediate nature have been found (e.g.
Ma et al. 2007,
their Table 3), raising the
question asked by Forbes & Kroupa for which, perhaps not
surprisingly, no clear answer has yet emerged. While those authors
explored the notion of a division by, among other properties, size and
luminosity, they did not discuss how the density varies. As an addendum
of sorts to
Forbes & Kroupa
(2011),
the density of elliptically-shaped objects is presented here in
Figure 1b. This is also done to allow the author
to wave the following flag.

Apparent in Figure 1b, but apparently not well
recognised within the community, is that the bulges of disc galaxies can
be much denser than elliptical galaxies. If the common idea of galaxy
growth via the accretion of a disc, perhaps from
cold-mode accretion streams, around a pre-existing spheroid is correct (e.g.
Navarro & Benz 1991;
Steinmetz & Navarro
2002;
Birnboim & Dekel
2003;
see also
Conselice et al. 2011
and
Pichon et al. 2011),
then one should expect to find dense spheroids at high-z with
1010 - 1011M
of stellar material, possibly surrounded by a faint (exponential) disc
which is under development. It is noted here that the dense, compact
early-type galaxies recently found at redshifts of 1.4-2.5
(Daddi et al. 2005;
Trujillo et al. 2006)
display substantial overlap with the location of present day bulges in
Figure 1a, and that the merger scenarios for
converting these compact high-z galaxies into today's elliptical
galaxies are not without problems (e.g.
Nipoti et al. 2009;
Nair et al. 2011).
It is also noted that well-developed discs and disc galaxies are rare at
the redshifts where these compact objects have been observed alongside
normal-sized elliptical galaxies.
Before trying to understand galaxy structure at high-redshift, and galaxy
evolution - themes not detailed in this review - it is important to
first appreciate galaxy structures at z = 0 where observations
are easier and local benchmark scaling relations have been established.

1 This review
does not encompass dwarf spheroidals, or any, galaxies fainter than
MB -14 mag. These
galaxies can not be observed (to date) at cosmologically
interesting distances, and their increased scatter in the colour-magnitude
relation may indicate a range of galaxy types (e.g.
Penny & Conselice
2008,
and references therein).
Back.

3 The NGC built upon the (Herschel
family's) Catalog of Nebulae and Clusters of Stars
(Herschel 1864).
Back.

4Reynolds (1927)
called Hubble's attention to pre-existing and
partly-similar galaxy classification schemes which were not cited.
Back.

5 It is of interest to note that
Hubble (1934,
1936b,
1937)
was actually cautious to accept that the redshifts corresponded to real
velocities and thus an expanding Universe as first suggested by others.
He used the term "apparent velocity" to flag his skepticism.
In point of fact,
Hubble & Tolman
(1935)
wrote that the data is "not yet sufficient to permit a decision between
recessional or other causes for the red-shift".
Back.